Article

# On a Vizing-like conjecture for direct product graphs.

Department of Mathematics, PEF, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia

Discrete Mathematics (Impact Factor: 0.58). 01/1996; 156:243-246. DOI: 10.1016/0012-365X(96)00032-5 Source: DBLP

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**ABSTRACT:**Let $G_1$ and $G_2$ be two undirected nontrivial graphs. The Kronecker product of $G_1$ and $G_2$ denoted by $G_1\otimes G_2$ with vertex set $V(G_1)\times V(G_2)$, two vertices $x_1x_2$ and $y_1y_2$ are adjacent if and only if $(x_1,y_1)\in E(G_1)$ and $(x_2,y_2)\in E(G_2)$. This paper presents a formula for computing the diameter of $G_1\otimes G_2$ by means of the diameters and primitive exponents of factor graphs.Mathematical Sciences Letters. 04/2013; 2(2):121-127. - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper first presents some earlier results on k-dominating sets of Cartesian products and cardinal products of two paths, and then new results on k-dominating sets of a strong product and an equivalent product of two paths (a k-domination number of a strong and an equivalent product of Pm × Pn for arbitrary m and n).Croatica Chemica Acta (CCA@chem.pmf.hr); Vol.80 No.2. 01/2007; -
##### Article: Lower bounds for the domination number and the total domination number of direct product graphs

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**ABSTRACT:**A sharp lower bound for the domination number and the total domination number of the direct product of finitely many complete graphs is given: γ(×i=1tKni)≥t+1,t≥3. Sharpness is established in the case when the factors are large enough in comparison to the number of factors. The main result gives a lower bound for the domination (and the total domination) number of the direct product of two arbitrary graphs: γ(G×H)≥γ(G)+γ(H)−1γ(G×H)≥γ(G)+γ(H)−1. Infinite families of graphs that attain the bound are presented. For these graphs it also holds that γt(G×H)=γ(G)+γ(H)−1γt(G×H)=γ(G)+γ(H)−1. Some additional parallels with the total domination number are made.Discrete Mathematics. 12/2010; 310(23):3310–3317.

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